International Fixed Income

Topic IA: Fixed Income Basics-Valuation

January 2000

Readings

• Overview of Forward Rate Analysis (Ilmanen, Salomon Brothers)

Outline

I. Bond pricing basics

A. The discount function

B. Valuing fixed cash flows

C. Yields

D. Forward rates

E. Term structure theories

Discount Factors

The most basic debt instrument is a zero-coupon bond. Because of the time value of money, a dollar today is worth more than a dollar received in the future, so the price of a zero is always less than its face value. Let

dt

denote the t-period discount factor, i.e., the price today of an asset that pays $1 in t periods.

Discount Factors: An Example

US Strips Market

6-month 97.30

1-year 94.76

1.5-year 92.22

2-year 89.72

5-year 75.41

The Discount Function

• The Discount Function gives the discount factor or unit zero price as a function of maturity.

• Because of the time value of money, longer zeroes have lower prices. Therefore, the discount function is always downward-sloping.

The Discount Function: An Example

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Years to Maturity

d t

US Strips Market

Spot Rates of Interest

Spot rate: the t-year spot rate is the semiannually compounded rate of return implied by the market price of the t-year zero.

Discount factors and spot rates of interest are closely related:

11

2

)2/1(

1

2/1

2

t

tt

tt

t

dr

rd

Spot Rates of Interest: An Example

US Strips Market

%73.512

%45.512

%54.512

10/17370.01

5

2/1

9476.01

1

1/19730.01

5.0

r

r

r

The Term Structure of Interest Rates

The relation between spot rates and time to maturity is called the term structure of interest rates (also known as the spot, yield, or zero curve).

The Spot Curve

5

5.5

6

6.5

7

0 5 10 15 20 25 30

Spot Cve

US Strips Market

Yield Curve Shapes

Upward Downward

Humped Flat

Spot Curve (1/9/98 - 1/7/00)

4

5

6

7

3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 15yr 20yr 30yr

1/9/19981/11/19991/7/2000

Outline

I. Bond pricing basics

A. The discount function

B. Valuing fixed cash flows

C. Yields

D. Forward rates

E. Term structure theories

A Coupon Bond as a Portfolio of Zeroes

6 months 1 year 1 1/2 years

$425 $425 $10425

Example: $10,000 par of a 1.5 year, 8.5% T-bondmakes the following payments:

This is the same as a porfolio of three zeroes:• $425 par of a 6-mth zero•$425 par of a 1-yr zero•$10,425 par of a 1.5 yr zero

The Principle of No Arbitrage or The Law of One Price

Two assets which offer exactly the same cash flows must sell for the same price.

Why? If not, then one could buy the cheaper asset and sell the more expensive, making a profit today with no cost in the future. This arbitrage opportunity cannot persist in equilibrium.

Valuation by Replication

The previous example shows that we can construct a coupon bond from zeros. Therefore, given a set of discount factors, we can value a coupon bond by valuing the portfolio of zeros that replicates its cash flows.

Valuing A 1.5-Year, 8.5% T-Note

5.115. 10425$425$425$ dddV

Time Cash Flow Discount Factor Value

0.5 $425 0.9730 $4141.0 $425 0.9476 $4031.5 $10425 0.9222 $9614

Total $10430

An Arbitrage Opportunity

• What if the 1.5-year 8.5% coupon bond were worth only 104% of par value?

• You could buy, say $1 million par of the bond for $1,040,000 and sell the cash flows off individually as zeroes for total proceeds of $1,043,000, making $3000 of riskless profit.

• Similarly, if the bond were worth more than 104.3% of par, you could buy the portfolio of zeroes, reconstitute them, and sell the bond for riskless profit.

The General Approach

Suppose we have an asset whose cash flows are risk-free.

Then, by no arbitrage, the market value of the asset must be:

)()()(

111

21

2

2

22

21

2

2

1

1

Nttt

Nr

t

r

t

r

t

CFdCFdCFd

CFCFCFV

N

Nt

N

tt

Application: Synthetic Bonds

• A synthetic bond is a portfolio of securities that is constructed to produce a specific pattern of cash flows

• Uses– exploit mispricing (conversion arbitrage)– hedge a series of future cash flows– price complex securities by replication

Outline

I. Bond pricing basics

A. The discount function

B. Valuing fixed cash flows

C. Yields

D. Forward rates

E. Term structure theories

Definition

• Definition: the yield-to-maturity of a bond is defined as the discount rate that makes the market price of the bond equal to the discounted value of its future cash flows.

• The yield-to-maturity is often called the internal rate of return, or the redemption yield.

Mathematical Definition

Mathematically, the yield-to-maturity is the interest rate (y) that solves the equation:

Ty

McM

y

cMy

cMP 22

21

)2/(

21

)2/(

21

)2/(

c is the annual coupon rateM is the face value of the bondP is the market price of the bond

Example

• Consider the aforementioned 8.5% 1.5-year T-bond. What’s its yield?

32

21

25.104

21

25.4

21

25.43.104

yyy

Solving for the yield, we get 5.47%

Valuation Formulas

We have expressions for the value of a portfolio of fixed cash flows in terms of– discount factors (by no arbitrage)– discount rates (by the definition of the discount

rates)– yield (by the definition of yield).

Valuation Formulas continued...

jtyj

jtjtrj

jj

N

jt

N

jt

N

jtt

C

C

dCV

2

2

2

2

)1(

1

1

)1(

1

1

1

Interpretation

321

321

)2/0547.01(

0425.1

)2/0547.01(

0425.0

)2/0547.01(

0425.00430.1

)2/0547.1(

0425.1

)2/0545.1(

0425.0

)2/0554.1(

0425.00430.1

Compare the formula with discount rates and yieldin our example:

Interpretation...

Yield to maturity is just a complex, nonlinear “average” of spot rates of interest.

– Because most of the bond’s cash flow arrives at maturity (the principal), the T-year spot rate gets the most weight in the yield-to-maturity calculation.

– High coupon bonds pay a larger percentage of their face value as coupons than low coupon bonds; thus, their yields-to-maturity give more weight to earlier spot rates.

Interpretation...

• The yield of a portfolio of fixed cash flows depends on the size and timing of those cash flows.

• The yield is more heavily influenced by cash flows that are – larger in size (in present value terms)– later in time (for a given present value)

Conclusions about Yield

• Yields are not necessarily a good measure of value:– When the term structure is not flat, bonds with

different cash flows should generally have different yields in the absence of arbitrage.

– This is true even if the bonds have the same maturity.

Upward Sloping Yield Curve

Maturity

Yield

High coupon

Zero curve

Low coupon

Downward Sloping Yield Curve

Maturity

Yield

High coupon

Zero curve

Low coupon

Outline

I. Bond pricing basics

A. The discount function

B. Valuing fixed cash flows

C. Yields

D. Forward rates

E. Term structure theories

Forward Contracts

• Forward contract: a binding agreement to buy or sell a fixed quantity of an asset at an agreed upon price (called the forward price) at a specific time in the future.

• The terms of the contract are set today.

• No money exchanges hands today. Delivery and settlement occur on the future date.

Forward Loan

• One way to think of a forward purchase of a zero is as a forward loan.

• You contract today to lend $F at date t and be repaid $1 at date T.

0

-

t

$1

T

FtT

Intuition

• Going long a T-period zero

• Financed by going short a t-period zero

This is equivalent to

No money exchanges hands today (as true ofall forward contracts) since your short positionin the t-year zero pays for your current purchaseof the T-year zero.

Forward Rates

• Basic concept: a forward contract on a zero-coupon bond is a binding agreement to make or take a loan on a future date. The loan principal is the face value of the zero, so the forward price determines the rate of interest on the loan. This rate is called the forward rate of interest.

• Definition: the forward rate of interest for the period t to t+T is the rate of interest available today for a loan which goes from date t (the expiration date of the forward contract) to date t+T (the maturity date of the loan).

Forward Rate

• People try to summarize the terms of the forward loan by quoting the forward rate.

• The annualized, semi-annually compounded forward rate f is defined by:

)(2)2/1(

1tTT

t

Tt f

F

Forward Rate as Discount rates:PROOF

d

dt

T

( / ) ( )1 212 f

FtT T t

tT

( / )

( / )

1 2

1 2

2

2

r

rT

T

tt

Forward Rate Diagram

00 Tt

( / )1 2 2 rTT

( / )1 2 2 rtt ( / ) ( )1 2 2 f t

T T t

( / )

( / )

1 2

1 2

2

2

r

rT

T

tt

( / ) ( )1 2 2 f tT T t

Forward Rate Example

%36.5

)2/0554.1(

)2/0545.1()2/1(

)2/1(

)2/1()2/1(

15.0

215.0

5.025.0

121)5.01(21

5.0

f

f

r

rf

What is the semi-annual compounded forwardfor a six-month loan starting in six months?

Spot Rates as Averages of Forward Rates

d d F F Ft tt 0 5 0 5

111 5

0 5. ..

.

( / ) ( / ) ( / ) ( / ). . .1 2 1 2 1 2 1 220 5 0 5

10 5r r f ft

ttt

The discount rate is the geometric average of allthe forward rates. In terms of our example, the6-month spot rate is 5.54%, and the forward rateis 5.36%. The average is equal to the one-year rate of 5.45%.

Interpretation

• The forward rate is the marginal rate for extending the length of the loan.– In the above example, the marginal rate from

investing in 1-year strips instead of 6-month strips is 5.36%, which is below the current 5.54% 6-month rate.

The Forward Curve

• Generally, when someone uses the term “forward curve”, they are referring to the set of 1-period implied forward rates today as a function of time.

• The T-year forward curve is composed of 2(T-1) forward rates: the rate from year .5 to 1, the rate from year 1 to 1.5, ..., the rate from year T-.5 to T.

The Forward Curve

4

4.5

5

5.5

6

6.5

7

7.5

8

0 5 10 15 20 25 30

Time to Maturity

Yield

Spot

Forward

US Spot and Forward Curve

Practical Consideration

• One problem with the forward curve estimated via the strip rates is that it is too jagged.– errors in observable prices (bid/ask spread)

– only finite number of bonds

– liquidity of bonds

• Alternative method in practice is to use spline estimation, which takes a limited number of bonds and fits a smooth curve to estimate the discount function.

The Forward Curve

4

4.5

5

5.5

6

6.5

7

7.5

8

0 5 10 15 20 25 30

Time to Maturity

Yield

SpotForward

US Spot and Forward Curve

Outline

I. Bond pricing basics

A. The discount function

B. Valuing fixed cash flows

C. Yields

D. Forward rates

E. Term structure theories

The Expectations Theory

Expectations Theory: implied forward rates are unbiased forecasts of future spot rates.

Note that nominal spot rates are a function of inflation and real interest rates.

Implications for the Yield Curve

Under the expectations theory– an upward sloping yield curve signals that

interest rates are expected to rise in the future– a flat yield curve signals that they are expected

to stay the same– a downward sloping yield curve signals that

interest rates are expected to fall

Implied Spot Curve Curve for 1/16/2001

5.5

6

6.5

7

7.5

3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr

1/16/2001

Implied Spot Curve Curve for 1/11/2000Based on Forward Curve 1/11/1999

4

5

6

7

3mo 6mo 1yr 2yr 3yr 5yr 7yr 10yr 20yr

Actual SpotImplied Spot

Empirical Evidence

• Historically, the yield curve has been upward sloping on average.

• If the expectations theory is true, then the market is forecasting an increase in interest rates most of the time. This is not consistent with rational expectations.

The Liquidity Preference Hypothesis

Liquidity preference hypothesis: For liquidity reasons, investors may have to sell a bond before it reaches maturity. Thus, they face less price risk (interest rate risk) if they invest in short-term instruments.

Risk Premiums

Under the liquidity preference theory, the expected future spot rate and the forward rate differ by a liquidity premium

f E r L1 2 1 2 1 2, , ,[ ]

Implications for the Yield Curve

Under the liquidity preference theory– an upward sloping yield curve is ambiguous– a flat yield curve signals that interest rates are

expected to fall– a downward sloping yield curve signals that

interest rates are expected to fall substantially

Empirical Evidence

• Average holding-period returns for bonds increase with maturity. This supports the hypothesis that investors have liquidity preferences.

• But premiums appear to move substantially over time.